Relative tensor triangular Chow groups, singular varieties and localization

We extend the scope of Balmer’s tensor triangular Chow groups to compactly generated triangulated categories $\mathcal{K}$ that only admit an action by a compactly-rigidly generated tensor triangulated category $\mathcal{T}$ as opposed to having a compatible monoidal structure themselves. The additional flexibility allows us to recover the Chow groups of a possibly singular algebraic variety $X$ from the homotopy category of quasi-coherent injective sheaves on $X$. We are also able to construct localization sequences associated to restricting to an open subset of $\mathrm{S}\mathrm{p}\mathrm{c}({\mathcal{T}}^c)$, the Balmer spectrum of the subcategory of compact objects $$\mathcal{T}$^c \subset $\mathcal{T}$$. This should be viewed in analogy to the exact sequences for the cycle and Chow groups of an algebraic variety associated to the restriction to an open subset.